Klimeck – ECE606 Fall 2012 – notes adopted from Alam ECE606: Solid State Devices Lecture 23 MOSFET I-V Characteristics MOSFET non-idealities Gerhard Klimeck [email protected]Klimeck – ECE606 Fall 2012 – notes adopted from Alam Outline 2 1) Square law/ simplified bulk charge theory 2) Velocity saturation in simplified theory 3) Few comments about bulk charge theory, small transistors 4) Flat band voltage - What is it and how to measure it? 5) Threshold voltage shift due to trapped charges 6) Conclusion Ref: Sec. 16.4 of SDF Chapter 18, SDF
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Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Outline
2
1) Square law/ simplified bulk charge theory2) Velocity saturation in simplified theory3) Few comments about bulk charge theory, small
transistors 4) Flat band voltage - What is it and how to measure it?5) Threshold voltage shift due to trapped charges6) Conclusion
Ref: Sec. 16.4 of SDF Chapter 18, SDF
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Post-Threshold MOS Current (VG>Vth)
3
( )0
= − ∫DSV
D effch
iQI VVW
dL
µ
1) Square Law
2) Bulk Charge
3) Simplified Bulk Charge
4) “Exact” (Pao-Sah or Pierret-Shields)
[ ]) )( = − − −i G G TC V VQ mVV
[ ]) )( = − − −G Ti GC V VV VQ
( )2 22( )
+ = − − − − −
Si A BG Gi FB B
O
Qq N V
C V V VVC
ε φψ
Formula overview –derivation to follow
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
2ψ B
Effect of Gate Bias
4
P
S G D
B
VGS
n+ n+
Gated doped or p-MOS with adjacent n+ regiona) gate biased at flat-bandb) gate biased in inversion
A. Grove, Physics of Semiconductor Devices, 1967.
WDM
VGS > VT
WD
VBI
y
No source-drain bias
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
The Effect of Drain Bias
Alam ECE-606 S09 5
2D band diagram for an n-MOSFET
a) device
b) equilibrium (flat band)
c) equilibrium (ψS > 0)
d) non-equilibrium with VG and VD >0 applied
SM. Sze, Physics of Semiconductor Devices, 1981 and Pao and Sah.
FNDepletion very different in source and drain side Gate voltage must ensure channel formation=> LARGE
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Effect of a Reverse Bias at Drain
6
ψ S = 2ψ B + VRVBI + VR
WDM
VGS > VT VR( )
WD
VRVR
Gated doped or p-MOS with adjacent, reverse-biased n+ regiona) gate biased at flat-bandb) gate biased in depletionc) gate biased in inversion
A. Grove, Physics of Semiconductor Devices, 1967.
VR
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Inversion Charge in the Channel
7
P
S G
B
n+ n+ ( )
( ( 0)( ) )i ox G th
TA T
Q C V V V
q VVN W W
= − − −+ − =
VG=0VD=0
VVB
VG=0VD>0
VD
VB
VG>0VD>0
D
Channel - Drain)Band diagrams (Gate –
Channel - Drain)
Body potential = constant, n+ region potential lowered
Due to drain bias, additional gate voltage (as compared to threshold in MOSCAP) is now needed to invert the channel throughout its
length
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Inversion Charge at one point in Channel
8
VGVD
VBVG
( )* 2( )A
th Fox
Tq W VNV V
Cφ= + −2
( 0)A Tth F
ox
Wq
C
VNV φ= =−
( )* ( ( 0))T TAth th
ox
qNV V
W WVV
V
C
=−= + −
*( )i ox G thQ C V V= − −
V
Threshold voltage in the presence of drain
bias
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Approximations for Inversion Charge
9
( )( ) ( ) ( 0)i O G th A T TQ C V V V qN W V W V= − − − + − =
( ) ( )2 2( ) 22 = − − − + −
+O G th S o A S oB A BC V V V q N q NVφ φκ ε κ ε
( )i ox G thQ C V V V≈ − − −
( )i ox G thQ C V VmV≈ − − −
Approximations:
Square law approximation …
Simplified bulk charge approximation …
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
The MOSFET
10
P
S G D
B
n+ n+
VS = 0 VD > 0
Fn = Fp = EF
Fn = Fp − qVD
Fn increasingly negative from source to drain(reverse bias increases from source to drain)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Elements of Square-law Theory
11
VG VD>00
y
GCA : Ey << Ex
[ ]( ) ( )i ox G thQ y C V V mV y= − − −
VG V VG
Voltage and hence inversion charge vary spatially
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Charge along the channel ….
p
n
( )− −= C nE FCn N e β ( )−= V pE F
Cp N e β
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Charge along the channel …
( )− −= C nE FCn N e β ( )−= V pE F
Cp N e β
VG VD>00
VD
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Depletion into the channel ….
NA
n
( )− −= C nE FCn N e β ( )−= V pE F
Cp N e β
VG
WT(VD=0) WT(VD)
VD
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Depletion into the channel …
VG VD>00
Depletion
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Another view of Channel Potential
16
FP
FNFN
FPFP
FN
EC
EFEV
N+ N+P-doped
Source Drain
x
x
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Square Law Theory
17
VG VD>00
VD
Q
V1, 1,
1, 0
( )D
ii
i N i N
V
Dox G th
i N
J dyQ dV
Jdy C V V mV dV
µ
µ
= =
=
=
= − −
∑ ∑
∑ ∫
1 1 1 1
1
2 2 2 2
2
3 3 3 3
3
4 4 4 4
4
dVJ Q Q
dy
dVJ Q Q
dy
dVJ Q Q
dy
dVJ Q Q
dy
µ µ
µ µ
µ µ
µ µ
= =
= =
= =
= =
E
E
E
E
( )2
2ox D
D G th Dch
C VJ V V V m
L
µ = − −
Q1Q2 Q3 Q4
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Square Law or Simplified Bulk Charge Theory
18
( )2
2o
D G Tch
W CI V V
mL
µ= −ID
VDS
VGS
( )D o G T D
WI C V V V
Lµ= −
VDSAT = VGS − VT( )/ m
( )2
2ox D
D G th Dch
C VI W V V V m
L
µ = − −
( )2
2ox D
D G th Dch
C VJ V V V m
L
µ = − −
( ) ( )*,0= = − − ⇒ = −D
G th D D sat G th
dIV V mV V V V m
dV
Region of validity for the expression for currents
Unphysical, since current does not decrease with increase in
Vd
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Why square law? And why does it become invalid
19
( )2
2o
D G Tch
W CI V V
mL
µ= −
ID
VDS
VGS
VDSAT = VGS − VT( )/ m
( )i ox G thQ C V V mV≈ − − −
VG VD>00
Q
This situation doesn’t arise since electrons travelling from left to right are swept into the drain under the effect of the reverse bias applied
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Linear Region (Low VDS)
20
ID
VDS small
VGS
( )D o G T Dch
DS
CH
WI C V V V
L
V
R
µ= −
=
Actual
SubthresholdConduction
VT
Mobility degradation at high VGS
Slope gives mobility
Intercept gives VT
Can get VT also from C-V
( )2
2ox D
D G th D
ch
C VI W V V V m
L
µ = − −
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Outline
21
1) Square law/ simplified bulk charge theory2) Velocity saturation in simplified theory3) Few comments about bulk charge theory, small
transistors 4) Flat band voltage - What is it and how to measure it?5) Threshold voltage shift due to trapped charges6) Conclusion
Ref: Sec. 16.4 of SDF Chapter 18, SDF
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Velocity vs. Field Characteristic (electrons)
Electric field V/cm --->
velo
city
cm
/s -
-->
107
104
υ = υ sat
1/ 221 ( )
−= +
d
c
µυ E
E E
1 ( )
−=+d
c
µυ E
E E
, =d sat cυ µEυ = µE
22
Velocity saturates at high fields because of scattering
This expression can be used to re-derive the expression for current since since mobility is now, in principle, a function of distance
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Recap - derivation for MOSFET current
23
VG VD>00
1 1 1 1 1 1
1
2 2 2 2 2 2
2
3 3 3 3 3 3
3
4 4 4 4 4 4
4
dVJ Q Q
dy
dVJ Q Q
dy
dVJ Q Q
dy
dVJ Q Q
dy
µ µ
µ µ
µ µ
µ µ
= =
= =
= =
= =
E
E
E
E1, 1,( )= =
⇒ =∑ ∑ii
i N i N
J dyQ dV
yµ
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Velocity Saturation
24
VD
Q
V
υ vsat
1, 00
( )
1=
= − −
+
∑ ∫DV
D ox G thi N
c
dyJ C V V mV dV
µE
E
( )2
0
2ox D
D G th DD
chc
C mVJ V V V
VL
µ = − −
+E
( )2
0 0 2
11
+ = − −
∫chL
D Dox G th D
c
J mVdy C V V V
dV
dyµ E
( )2
0 0 2
+ = − −
∫ ∫ch DSL
D
V
Dox G t D
chD
mVC
Jd VJ V V
EVdy
VG VD>00
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Significance of the new expression
25
( )2
0
2ox D
D G th DD
chc
C mVJ V V V
VL
µ = − −
+E
• At very small channel lengths and high drain biases, the current expression becomes independent of the channel length
• In the linear region in the I-Vd characteristics, you have a resistance that doesn’t depend on the length of the channel
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Calculating VDSAT
dID
dVDS
= 0
( )2
2
= − −
+
D o ox DG th D
Dch
c
I C VV V V m
VW L
µ
E
VDSAT
=2 V
G−V
th( ) / m
1+ 1+ 2µo
VG
−Vth( ) mυ
satL
ch
26
Take log on both sides and then set the derivative to zero ….
<V
GS−V
T( )m
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Velocity Saturation in short channel devices
( )= −D ox sat G TI WC V VυID
VDS
VGS
( )2
,
,0, , 2
= − −
+
D satoxD sat G th D sat
cD sat
Ch
mVCJ V V
LV
Vµ
E
27
This expression can be derived by plugging in the value of Vd,sat for the short channel regime
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
‘Linear Law’ Expression at the limit of L --> 0
( )( )0
2 /
1 1 2
−=
+ + −G th
DSAT
G th sat ch
V V mV
V V m Lµ υ
( )= −DSAT ox sat G thI W C V Vυ
( ) 02→ −DSAT sat ch G thV L V V mυ µ
( ) ( )( )
0
0
1 2 1
1 2 1
+ − −= −
+ − +G th sat ch
DSAT ox sat G th
G th sat ch
V V m LI W C V V
V V m L
µ υυ
µ υ
Complete velocity saturation
Current independent of L
28
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
‘Signature’ of Velocity Saturation
29
ID
VDS
VGS
( )−=D sa G ht toxI W C V Vυ
ID
VDS
VGS
( )0
2
2
−=D ox
ch
G thWI C
V V
L mµ
Can pull out oxide thickness from experimental curves… How?
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
ID and (VGS - VT): In practice …..
30
ID
VDS
VGS
( )( ) ~= −D D DD G thI V V V Vα
1< α < 2
Long channelComplete velocity saturation
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Outline
31
1) Square law/ simplified bulk charge theory2) Velocity saturation in simplified theory3) Few comments about bulk charge theory, small
transistors 4) Flat band voltage - What is it and how to measure it?5) Threshold voltage shift due to trapped charges6) Conclusion
Ref: Sec. 16.4 of SDF Chapter 18, SDF
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Approximations for Inversion Charge
( )( ) ( ) ( 0)i O G th A T TQ C V V V qN W V W V= − − − + − =
( ) ( )( ) 2 2 2 2O G th S o A B S o A BC V V V q N V q Nκ ε φ κ ε φ= − − − + + −
( )i ox G thQ C V V V≈ − − −
( )i ox G thQ C V V mV≈ − − −
Approximations:
Square law approximation …
Simplified bulk charge approximation …
32
One could substitute the expression for Qi above explicitly instead of using m to simplify the equation, resulting in a more complete bulk charge expression
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Complete Bulk-charge Theory
01,0 0
[.........( ) .]=
= − − +∑ ∫ ∫D DV
DO G th
i N
VJ
dy C V V V dV dVµ
( )3 / 22
0 4 31 1
3 2 42
+ − +
= − − −
A T D DF
O F
ox DD G th D
ch F
C VJ V V V
L
qN W V V
Cφ
φµ
φ
00 0 0
[...... . .]( ) . .= − − +∫ ∫ ∫ch D DL V
DO G th
VJ
dy C V V V dV dVµ
(Eq. 17.28 in SDF) …. Explicit dependence on bulk doping
33
Additional V dependent terms abstracted into m previously
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Velocity Overshoot
0.0 10 0
5.0 10 6
1.0 10 7
1.5 10 7
2.0 10 7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.5 1 1.5
Ave
rage
vel
ocity
(cm
/s)
Kin
etic
ene
rgy
per
elec
tron
(eV
)
Position (µm)
103 V/cm 103 V/cm105 V/cm
υ ≠ µn E( )E
υsat
34
Valid for bulk semiconductors, not valid at top of the barrier
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Velocity Overshoot in a MOSFET
35
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Intermediate Summary
1) Velocity saturation is an important consideration for short
depends on substrate (bulk) doping. In the simplified bulk
charge theory, doping dependence is encapsulated in m.
3) Additional considerations of velocity overshoot could
complicate calculation of current.
4) Good news is that for very short channel transistors,
electrons travel from source to drain without scattering. A
considerably simpler ‘Lundstrom theory of MOSFET’
applies. 36
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Outline
37
1) Square law/ simplified bulk charge theory2) Velocity saturation in simplified theory3) Few comments about bulk charge theory, small
transistors 4) Flat band voltage - What is it and how to measure
it?5) Threshold voltage shift due to trapped charges6) Conclusion
Ref: Sec. 16.4 of SDF Chapter 18, SDF
,
( )IT sM M Fth th ideal
O OMS
O
QQ QV V
C C C
φγφ= + − − −
( )( ) ~= −D D DD G thI V V V Vα
1< α < 2
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
(1) Idealized MOS Capacitor
38
Substrate (p)yχs
Φm
χi
EC
EVEF
p semiconductormetal insulator
Vacuum level
,( )= −i ox G th idealQ C V V
Recall that
,
2=
= −s F
Bth ideal s
ox
QV
C ψ φ
ψ
In the idealized MOS capacitor, the Fermi Levels in metal and semiconductor align perfectly so that at zero applied bias, the energy bands are flat
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Potential, Field, Charges
39
χs
Φm
χiV
E
Vbi=0 ρ
x
x
x
No built in potential, fields or charges at zero applied bias in the idealized MOS structure
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Real MOS Capacitor with ΦΦΦΦM < ΦΦΦΦS
40
ΦM = qφm χS
ΦS
qψ S > 0
EC
EV
EF
EVAC
EC
EV
EF
Note the difference
Do we need to apply less or more VG to invert the channel ?
In reality, the metal and semiconductor Fermi Levels are never aligned perfectly � when you
bring them together there is charge transfer from the bulk of the semiconductor to the
surface so that we have alignment
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Physical Interpretation of Flatband Voltage
41
ψS = 0 flat band
EC
EV
EF
0FB ms biV Vφ= = − <
VG = VFB < 0
EC
EV
EF
VG = 0
Vbi = −φms > 0+ −
The Flatband Voltage is the voltage applied to the gate that gives zero-band bending in the MOS structure. Applying this voltage nullifies the effect of the built-in potential. This voltage needs to be
incorporated into the idealized MOS analysis while calculating threshold voltage
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
How to Calculate Built-in or Flat-band Voltage
42
χs
Φm
Vacuum level
EV
EF
EC
( )= + − ∆ −
=
Φ
≡bi g p
FB MS
s MqV E
qV φ
χ qVbi
( )i ox G thQ C V V= −
Therefore,
2
= − −
Bth F
oxFB
QV
CVφ
The presence of a flatband voltage lowers or raises the threshold voltage of a MOS structure. Engineering question � Is it desirable to have a metal having a work function greater or less
than the electron affinity+(Ec-Ef) in the semiconductor?
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Measure of Flat-band shift from C-V Characteristics
43
C/Cox
VG
Ideal Vth
Actual Vth
The transition point between accumulation and depletion in a non-ideal MOS structure is shifted to the left when the metal work function is smaller that the electron affinity +(Ec-Ef). At zero applied bias the semiconductor is already depleted so that a very small positive bias inverts the channel. The flatband voltage is the amount of voltage required to shift the
curve such that the transition point is at zero bias.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Outline
44
1) Square law/ simplified bulk charge theory2) Velocity saturation in simplified theory3) Few comments about bulk charge theory, small
transistors 4) Flat band voltage - What is it and how to measure it?5) Threshold voltage shift due to trapped charges6) Conclusion
Ref: Sec. 16.4 of SDF Chapter 18, SDF
,
( )= + − −− M M F IT st
o
h th ideal M
x
S
ox ox
Q QV
CCV
C
Q φγφ
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
(2) Idealized MOS Capacitor
45
χsΦm
χi
EC
EV
EF
p semiconductormetal insulator
Vacuum level
,( )= −i ox G th idealQ C V V
Recall that
,
2=
= −s F
Bth ideal s
ox
QV
C ψ φ
ψ
Substrate (p)y
Qox
=0
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Distributed Trapped charge in the Oxide
46
EC
EV
EF
Ox
0x
= − −F Mth S
oM
x ox
Q QV
C Cψ γ
0
( )OX
M ox xQ dxρ= ∫
0
0
0
00
0
( )
( )
≡ =∫
∫
ox
ox
x
MM
x
x dxx
xxx dx
xρ
ργ
In the absence of charges in the oxide, the field is constant (dV/dx = constant). The presence of a charge distribution inside the oxide changes the field inside the oxide and effectively traps field lines comping from the gate. As a result, depending on the polarity of charges in the oxie, the threshold voltage is modified.
xm represents the centroid of the charge distribution – one can think of this as replacing the entire distribution with a
delta charge at this point
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
An Intuitive View
47
Ideal charge-free oxide
-E
0
Bulk charge
-E
0
-E
Interface charge
0
Reduced gate charge
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Gate Voltage and Oxide Charge
48
VG
=Vox
+ψs
2
20
( )− = =ox o
o
xx o
x
d V d
dx d
x
x κρ
εE
0 0( )
0( )
( ') 'x x
oxox
oxx x
x dxd
ρκ ε
=∫ ∫E
E
E
−dV
ox
dx=E
ox(x
0) −E
ox(x ) =E
ox(x
0) −
ρox
(x ')dx '
κox
ε00
x
∫
Vox
=κ
S
κox
x0E
S(x
0) − dx
0
x0
∫ρ
ox(x ')dx '
κox
ε00
x0
∫
=κ
S
κox
x0E
S(x
0) −
x ρox
(x )dx
κox
ε00
x0
∫
-E
0
Kirchoff’s Law – balancing voltages
Known from boundary conditions in semiconductor
and continuity of E
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Gate Voltage and Oxide Charge
49
0
0 00 0
(
( 2 )
1( 2 ) )) (
= = + ∆
= = + − ∫
th s F ox
x
Sos F xS
ox o
V V
x x x dxC x
x
ψ φ
κψ φκ
ρE
0
0
0
0
0
00
( )( )
x
Sox S
ox
x
o
o xx dxV x
xx
xκ ε
κκ
ρ∆ = −
∫E
0
0 00 0
( () )1= − ∫
oox
x
SS
ox x
x x x dxx
xCκ
ρκE
0
,0 0
,
( )1= −
= −
∫ o
x
th ideal
ox
Mth ideal
ox
x
M
V x dxC x
QV
C
xρ
γ
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Interpretation for Bulk Charge
50
0
1,0
1,
0
1
0
( ) ( )
(
1
)
= −
= −
−∫x
th th idealo
th id
o
o
x
Meal
V V x x x dx
Q x
xC x
xV
x C
ρ δ
C/Cox
VG
Ideal VT
New VT
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Interpretation for Interface Charge
51
0
*
0 0
*
( ) ( )1
ox o
x
th th
o
Fth
o
V V x dxC x
V
x x
C
x
Q
ρ δ= −
= −
−∫
C/Cox
VG
Ideal VT
New VT
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Time-dependent shift of Trapped Charge
52
E
0
, 10 0
1,
0
1( ) ( ( ))
( ) ( )
= − × −
= − ×
∫x
th th ideal oxox
oxth ideal
ox
V V xQ x x x t dxC x
x t Q xV
x C
δ
Sodium related bias temperature instability (BTI) issue
C/Cox
VG
Ideal VT
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Bias Temperature Instability (Experiment)
53
----------
+++++
+++++
M O S
(-) biases
0 xo
0.1xo
x
ρio
n
M O S----------
+++
++
+++++
(+) biases
x
0 xo
ρio
n
0.9xo
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Conclusion
54
1) Non-ideal threshold characteristics are important
consideration of MOSFET design.
2) The non-idealities arise from differences in gate and
substrate work function, trapped charges, interface
states.
3) Although nonindeal effects often arise from transistor